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Constructing Combinatorial Operads from Monoids

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Constructing Combinatorial Operads from Monoids FPSAC 2012, Nagoya, Japan DMTCS proc. AR, 2012, 229–240 Constructing combinatorial operads from monoids Samuele Giraudo1 1Institut Gaspard Monge, Universite´ Paris-Est Marne-la-Vallee,´ 5 Boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallee´ cedex 2, France Abstract. We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schroder¨ trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad. Resum´ e.´ Nous introduisons une construction fonctorielle qui, a` partir d’un mono¨ıde, produit une operade´ ensembliste. Nous obtenons de nouvelles operades´ (symetriques´ ou non) comme sous-operades´ ou quotients de l’operade´ obtenue a` partir du mono¨ıde additif. Celles-ci mettent en jeu divers objets combinatoires familiers : fonctions de parking, mots tasses,´ arbres plans enracines,´ chemins de Dyck gen´ eralis´ es,´ arbres de Schroder,¨ chemins de Motzkin, compositions d’entiers, animaux diriges,´ etc. Nous retrouvons egalement´ des operades´ dej´ a` connues : l’operade´ magmatique, l’operade´ commutative associative et l’operade´ diassociative. Keywords: Operad; Monoid; Generalized Dyck path; Tree; Directed animal; Diassociative operad. 1 Introduction Operads are algebraic structures introduced in the 1970s by Boardman and Vogt [1] and by May [13] in the context of algebraic topology. Informally, an operad is a structure containing operators with n inputs and 1 output, for all positive integer n. Two operators x and y can be composed at ith position by grafting the output of y on the ith input of x. The new operator thus obtained is denoted by x ◦i y. In an operad, one can also switch the inputs of an operator x by letting a permutation σ act to obtain a new operator denoted by x · σ. One of the main relishes of operads comes from the fact that they offer a general theory to study in an unifying way different types of algebras, such as associative algebras and Lie algebras. In recent years, the importance of operads in combinatorics has continued to increase and several new operads were defined on combinatorial objects (see e.g., [3, 4, 9, 10]). The structure thereby added on combinatorial families enables to see these in a new light and offers original ways to solve some com- binatorial problems. For example, the dendriform operad [10] is an operad on binary trees and plays an interesting role for the understanding of the Hopf algebra of Loday-Ronco of binary trees [8,11]. Besides, this operad is a key ingredient for the enumeration of intervals of the Tamari lattice [2, 3]. There is also a very rich link connecting combinatorial Hopf algebra theory and operad theory: various constructions produce combinatorial Hopf algebras from operads [5, 12]. 1365–8050 c 2012 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France 230 Samuele Giraudo In this paper, we propose a new generic method to build combinatorial operads. The starting point is to pick a monoid M. We then consider the set of words whose letters are elements of M. The arity of such words are their length, the composition of two words is expressed from the product of M, and permutations act on words by permuting letters. In this way, we associate to any monoid M an operad de- noted by TM. This construction is rich from a combinatorial point of view since it allows, by considering suboperads and quotients of TM, to get new operads on various combinatorial objects. This paper is organized as follows. In Section 2, we recall briefly the basics about set-operads. Section 3 is devoted to the definition of the construction associating an operad to a monoid and to establish its first properties. We show that this construction is a functor from the category of monoids to the category of operads that respects injections and surjections. Finally we apply this construction in Section 4 on various monoids and obtain several new combinatorial (symmetric or not) operads on the following combinatorial objects: endofunctions, parking functions, packed words, permutations, planar rooted trees, generalized Dyck paths, Schroder¨ trees, Motzkin paths, integer compositions, directed animals, and segmented integer compositions. We conclude by building an operad isomorphic to the diassociative operad [10]. Acknowledgements The author would like to thank Florent Hivert and Jean-Christophe Novelli for their advice during the preparation of this paper. This work is based on computer exploration and the author used, for this purpose, the open-source mathematical software Sage [15] and one of its extensions, Sage-Combinat [14]. 2 Preliminaries and notations 2.1 Permutations Let us denote by [n] the set f1; : : : ; ng and by Sn the set of permutations of [n]. Let σ 2 Sn, ν 2 Sm, 0 0 00 00 0 0 and i 2 [n]. The substitution of ν into σ is the permutation Bi(σ; ν) := σ1 : : : σi−1ν1 : : : νmσi+1 : : : σn 0 0 00 where σj := σj if σj < σi and σj := σj + m − 1 otherwise, and νj := νj + σi − 1. For instance, one has B4(7415623; 231) = 941675823. 2.2 Operads U Recall that a set-operad, or an operad for short, is a set P := n≥1 P(n) together with substitution maps ◦i : P(n) × P(m) !P(n + m − 1); n; m ≥ 1; i 2 [n]; (1) a distinguished element 1 2 P(1), the unit of P, and a symmetric group action · : P(n) × Sn !P(n); n ≥ 1: (2) The above data has to satisfy the following relations: (x ◦i y) ◦i+j−1 z = x ◦i (y ◦j z); x 2 P(n); y 2 P(m); z 2 P(k); i 2 [n]; j 2 [m]; (3) (x ◦i y) ◦j+m−1 z = (x ◦j z) ◦i y; x 2 P(n); y 2 P(m); z 2 P(k); 1 ≤ i < j ≤ n; (4) 1 ◦1 x = x = x ◦i 1; x 2 P(n); i 2 [n]; (5) (x · σ) ◦i (y · ν) = (x ◦σi y) · Bi(σ; ν); x 2 P(n); y 2 P(m); σ 2 Sn; ν 2 Sm; i 2 [n]: (6) Constructing combinatorial operads from monoids 231 The arity of an element x of P(n) is n. Let Q be an operad. A map φ : P!Q is an operad morphism if it commutes with substitution maps and symmetric group action and maps elements of arity n of P to elements of arity n of Q.A non-symmetric operad is an operad without symmetric group action. The above definitions also work when P is a N-graded vector space. In this case, the substitution maps ◦i are linear maps, and the symmetric group action is linear on the left. 3 A combinatorial functor from monoids to operads 3.1 The construction 3.1.1 Monoids to operads U Let (M; •; 1) be a monoid. Let us denote by TM the set TM := n≥1 TM(n), where for all n ≥ 1, TM(n) := f(x1; : : : ; xn): xi 2 M for all i 2 [n]g : (7) We endow the set TM with maps ◦i : TM(n) × TM(m) ! TM(n + m − 1); n; m ≥ 1; i 2 [n]; (8) defined as follows: For all x 2 TM(n), y 2 TM(m), and i 2 [n], we set x ◦i y := (x1; : : : ; xi−1; xi • y1; : : : ; xi • ym; xi+1; : : : ; xn): (9) Let us also set 1 := (1) as a distinguished element of TM(1). We endow finally each set TM(n) with a right action of the symmetric group · : TM(n) × Sn ! TM(n); n ≥ 1; (10) defined as follows: For all x 2 TM(n) and σ 2 Sn, we set x · σ := (xσ1 ; : : : ; xσn ) : (11) The elements of TM are words over M regarded as an alphabet. The arity of an element x of TM(n), denoted by jxj, is n. For the sake of readability, we shall denote in some cases an element (x1; : : : ; xn) of TM(n) by its word notation x1 : : : xn. Proposition 3.1 If M is a monoid, then TM is a set-operad. Proof: Let us respectively denote by • and 1 the product and the unit of M. First of all, thanks to (8) and (9), the maps ◦i are well-defined and are substitution maps of operads. Let us now show that TM satisfies (3), (4), (5), and (6). Let x 2 TM(n), y 2 TM(m), z 2 TM(k), i 2 [n], and j 2 [m]. We have, using associativity of •, (x ◦i y) ◦i+j−1 z = (x1; : : : ; xi−1; xi • y1; : : : ; xi • ym; xi+1; : : : ; xn) ◦i+j−1 z = (x1; : : : ; xi−1; xi • y1; : : : ; xi • yj−1; (xi • yj) • z1;:::; (xi • yj) • zk; xi • yj+1; : : : ; xi • ym; xi+1; : : : ; xn) = (x1; : : : ; xi−1; xi • y1; : : : ; xi • yj−1; xi • (yj • z1); : : : ; xi • (yj • zk); (12) xi • yj+1; : : : ; xi • ym; xi+1; : : : ; xn) = x ◦i (y1; : : : ; yj−1; yj • z1; : : : ; yj • zk; yj+1; : : : ; ym) = x ◦i (y ◦j z); 232 Samuele Giraudo showing that ◦i satisfies (3). Let x 2 TM(n), y 2 TM(m), z 2 TM(k), and i < j 2 [n]. We have, (x ◦j z) ◦i y = (x1; : : : ; xj−1; xj • z1; : : : ; xj • zk; xj+1; : : : ; xn) ◦i y; = (x1; : : : ; xi−1; xi • y1; : : : ; xi • ym; xi+1; : : : ; xj−1; xj • z1; : : : ; xj • zk; xj+1; : : : ; xn) (13) = (x1; : : : ; xi−1; xi • y1; : : : ; xi • ym; xi+1; : : : ; xn) ◦j+m−1 z = (x ◦i y) ◦j+m−1 z; showing that ◦i satisfies (4).
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